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A366073
The number of composite "Fermi-Dirac primes" (A082522) dividing n.
2
0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0
OFFSET
1,16
COMMENTS
First differs from A071325 at n = 36.
The number of "Fermi-Dirac primes" that are infinitary divisors of n is A064547(n).
LINKS
FORMULA
Additive with a(p^e) = floor(log_2(e)) = A000523(e).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} P(2^k) = 0.53331724743088069672..., where P(s) is the prime zeta function.
MATHEMATICA
f[p_, e_] := Floor[Log2[e]]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecsum(apply(exponent, factor(n)[, 2]));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 28 2023
STATUS
approved